ANZIAM J. 57 (EMAC2015) pp.C291–C304, 2016
C291
Mathematical fitting for the variation in
capacity of lithium iron phosphate batteries
corresponding to cycles
Chi-Yao Chung1
Zih-Yeh Lin4
Kung-Yen Lee2
Shuen-De Wu5
Ting-Jung Kuo3
Chiao Fu6
(Received 9 January 2016; revised 13 October 2016)
Abstract
The capacity of lithium iron phosphate (LiFePO4 ) batteries decreases as the usage cycles increase. In order to investigate the relationship between the number of charge/discharge cycles and the battery
capacity, two batteries were cyclically charged and discharged using an
automatic testing system, while simultaneously collecting the capacity
data. The tested batteries were composed of different materials and
had a capacity of 15 Ah. Each battery was cyclically charged and
discharged 800 times. The batteries are analyzed and compared with
each other to create fitted curves. The developed mathematical fitting,
c Austral. Mathematical Soc. 2016. Published
doi:10.21914/anziamj.v57i0.10436, October 26, 2016, as part of the Proceedings of the 12th Biennial Engineering Mathematics
and Applications Conference. issn 1445-8810. (Print two pages per sheet of paper.) Copies
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Contents
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which consists of both exponential and polynomial terms, is closely
responsive to the battery capacity.
Contents
1 Introduction
C292
2 Experiment
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3 Results and discussion
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4 Conclusion
C301
References
C302
1
Introduction
Lithium iron phosphate (LiFePO4 ) batteries have a variety of superior properties compared to more common lithium cobalt oxide batteries, such as higher
power densities, higher capacities, longer lifetimes and better safety. For these
reasons, LiFePO4 batteries are used extensively in electric vehicles, hybrid
electric vehicles, and energy storage devices [1, 2]. However, there is an issue
with the battery capacity in that it begins to rapidly decay after a certain
number of charge and discharge cycles, which may cause safety problems.
Therefore, it is very important to investigate the electrical characteristics
(voltage, current, and capacity) of LiFePO4 batteries in relation to the number
of cycles [3, 4, 5, 6, 7].
Because of electrochemical reactions and the aging of the material within the
battery, it is difficult to determine the remaining capacity of a battery with
respect to the number of cycles [3, 4, 8, 9]. Hence, the aim of this study is to
validate the proposed mathematical fitting by comparing measurement results
2 Experiment
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with fitted results. The proposed model contains both an exponential term
and a polynomial term. The parameters of the two terms are obtained by
fitting the capacity variation curve of the LiFePO4 batteries. The exponential
term represents the active electrochemical reaction during the initial usage
stage of a new battery. The polynomial term represents the steady state of the
battery. The capacity of the battery gradually decreases in the steady state.
2
Experiment
A model for LiFePO4 batteries is used to simulate the behavior of the batteries.
The battery model is an open-circuit voltage source connected to a single
series resistor and a single resistor-capacitor (rc) network. The capacity is
regarded as a nonlinear capacitor so that it simulates the capacity conditions.
According to Faraday’s Law, the battery capacity is equal to the integral of
current over time. In the charging and discharging experiments, the current
is constant so that the battery capacity is determined using the product of
the current and the time. Therefore, the variation in capacity after a single
charge/discharge cycle is obtained.
Figure 1 plots the experimental discharging voltage curves for new, 300cycle and 1000-cycle batteries for the constant-current (cc) and constantvoltage (cv) charging method and constant-current discharging method [10].
The figure indicates that the discharge time or the capacity of a battery
declines with an increasing number of cycles.
Two types of LiFePO4 batteries composed of different materials are measured
and compared. Each battery has the same capacity of 15 Ah as well as the
same number of cycles (800). At the beginning of the experiment, each battery
was charged using a dc power generator with the cc-cv method. The battery
was charged with the cc in the initial stage because a higher charging current
might cause permanent damage to the battery. When the battery voltage
reached the upper limit of the charging voltage, charging switched to the cv
2 Experiment
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Figure 1: The discharging voltage curves for new, 300-cycle, and 1000-cycle
batteries.
3 .4
V o lta g e ( V o lt)
3 .2
3 .0
2 .8
2 .6
2 .4
N e w
3 0 0 - C y c le
1 0 0 0 - C y c le
2 .2
2 .0
0
5 0 0
1 0 0 0
1 5 0 0
2 0 0 0
2 5 0 0
3 0 0 0
3 5 0 0
4 0 0 0
4 5 0 0
T im e ( s e c o n d s )
mode with a small current. This cc-cv method protects the battery from
damage and over-charging during the charge process. It also reduces the
charge time. The charging current rate and the upper limit charging voltage
were set at 1 C and 3.65 V, respectively. In the charge process, different
charging rates were not considered.
After the charging procedure, the battery was left idle until it was in a steady
state, and therefore there is no significant difference in the initial voltages of
the tested batteries, as shown in Figure 1. The batteries were then discharged
using a discharging current rate of 1 C through a dc electronic load. The 1 C
discharge rate means that it takes one hour to fully discharge a 15 Ah battery
with a discharging current of 15 A. The cut-off voltage was set at 2.0 V since
there is a significant difference in the lithium transfer reaction and irreversible
damage can occur when the voltage falls below 2.0 V [11, 12, 13, 14]. During
3 Results and discussion
C295
the discharging period, the battery voltage was automatically checked by a
computer. If the voltage reached the cut-off voltage, then the discharging
process was stopped. If not, then the process continued until the voltage
fell to 2.0 V. The capacity data was automatically recorded and collected
by a computer. The tested batteries were cyclically charged and discharged
800 times. Consequently, the capacity of all batteries is determined from the
integral of the discharging current over time.
A discharging current of 20 A was used in earlier experiments [10]. The
temperature and resistance increase with this higher current, and the battery
voltage decreases due to higher resistance, but the capacity of a battery
is almost the same [10]. For the discharging current of 20 A, the trend of
the capacity loss is almost the same as the curve shown in Figure 2 for the
discharging current of 15 A. However, the slope is steeper and the discharge
time is shorter for the 20 A case. If the temperature is too high, then the
capacity of a battery is reduced. This is because the higher thermal resistance
makes the voltage drop larger so that the battery voltage easily reaches the
cut-off voltage during the discharge process, and then the total discharge time
is shorter, leading to a smaller capacity.
3
Results and discussion
The experimental results and linear fitted curves for two types of battery,
Type-A and Type-B, are shown in Figure 2(a) and Figure 2(b), respectively.
The solid lines represent the experimental data and the dashed lines represent
the linear fitted curves. The results show that the battery capacities vary
depending on the type of battery. The capacities of the tested batteries reduce
as the number of cycles increases. To identify the best fitted equation for the
batteries, at first, two straight lines with negative slopes are programmed to
fit the real data for Type-A and Type-B batteries from about 100 cycles to
800 cycles. The linear fitted curves are then extrapolated to the initial point
(i.e., the number of cycles = 0). The divergences of the real data from the
3 Results and discussion
C296
Figure 2: The capacity variation and linear fitted curves for (a) Type-A and
(b) Type-B batteries.
1 5 .0
(a )T y p e -A
1 4 .8
L in e a r fitte d
1 4 .6
C a p a c ity (A h )
E x p e r im e n ta l
A c tiv e z o n e
( E x p o n e n tia l)
1 4 .4
1 4 .2
S te a d y Z o n e
( P o ly n o m ia l)
1 4 .0
1 3 .8
1 3 .6
1 3 .4
1 3 .2
1 3 .0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
N u m b e r s o f C y c le s
1 5 .0
(b )T y p e -B
1 4 .8
L in e a r fitte d
A c tiv e z o n e
( E x p o n e n tia l)
1 4 .6
C a p a c ity (A h )
E x p e r im e n ta l
1 4 .4
1 4 .2
1 4 .0
1 3 .8
1 3 .6
S te a d y Z o n e
( P o ly n o m ia l)
1 3 .4
1 3 .2
1 3 .0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
N u m b e r s o f C y c le s
6 0 0
7 0 0
8 0 0
3 Results and discussion
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linear fitted curves for Type-A and Type-B batteries are at cycles 100 and 150,
respectively (Figure 2). These divergences separate the experimental curves
into two parts. The first part is the active zone, during which the capacity is
unstable and decays exponentially; the second part is the steady zone, during
which there is a slow, linear decay in capacity. From Figure 2, we see that a
linear equation is unsuitable for the tested batteries when the capacity of the
battery is in the active zone.
From the experimental curves of Figure 2, the drop in capacity in the active
zone acts in a similar manner to the discharge behavior of an rc network [6].
Hence, the variation in capacity of the battery is a function of the number of
cycles and the fitted equation is
c(x) = ae−bx + sx + i ,
(1)
where x is the number of cycles, a and i are capacities, a + i is the initial
capacity (x = 0), b is the decay constant, and s is the linear decay rate in
the stable zone.
Equation (1) represents an equivalent electrical circuit which is composed
of a nonlinear open-circuit voltage source, a series resistor and one resistercapacitor (rc) network [10]. The nonlinear open-circuit voltage source represents the open voltage of a battery. In equation (1) ae−bx represents one
rc network because the charge and discharge behaviors of a capacitor is
described by an exponential term, and the sx + i term represents a linear
resistor.
Figure 3 shows there are no significant differences between the fitted curves
(dashed lines) and the experimental data curves (solid lines), indicating that
the proposed fitted equation (1) closely matches the individual data for these
two batteries. The exponential part dominates the rapid drop in capacity
when the battery is in the active zone. As the number of cycles increases, the
polynomial term dominates the decay rate of the capacity for the battery in
the steady zone.
Although Figure 4 only shows up to 800 cycles of charging and discharging, the
3 Results and discussion
C298
Figure 3: The fitted and experimental curves for (a) Type-A and (b) Type-B
batteries.
1 5 .0
(a )T y p e -A
1 4 .8
E x p e r im e n ta l
F itte d
C a p a c ity (A h )
1 4 .6
1 4 .4
1 4 .2
1 4 .0
1 3 .8
1 3 .6
1 3 .4
1 3 .2
1 3 .0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
N u m b e r s o f C y c le s
1 5 .0
(b )T y p e -B
1 4 .8
E x p e r im e n ta l
F itte d
C a p a c ity (A h )
1 4 .6
1 4 .4
1 4 .2
1 4 .0
1 3 .8
1 3 .6
1 3 .4
1 3 .2
1 3 .0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
N u m b e r s o f C y c le s
6 0 0
7 0 0
8 0 0
3 Results and discussion
C299
Table 1: The parameters of the mathematical fitting.
Batteries
Parameters
a (Ah) b (1/number) s (Ah/number) i (Ah)
Type-A
0.302
0.0319
−1.302 · 10−3
14.23
−3
Type-B
0.463
0.0254
−1.178 · 10
14.45
fitted curves should be feasible as long as the battery continues to operated in
the steady zone (e.g., up to 2000 cycles) thanks to the stability of an LiFePO4
battery.
Table 1 provides a summary of the parameters used in equation (1) for Type-A
and Type-B batteries. The values for the two types of batteries are similar.
The a and i values for the Type-B battery are larger than the a and i values
for the Type-A battery because the initial capacity of the Type-B battery is
larger than that of the Type-A battery. The value of b for the Type-B battery
is less than the value of b for the Type-A battery because the duration of the
active zone for the Type-B battery is longer than that of the Type-A battery.
The linear decay rates s are very close for both batteries.
Figure 4(a) shows the experimental data for both the Type-A and Type-B
batteries. The trends for the two types of battery are very similar. The
difference in capacity between the Type-A and Type-B batteries remains
almost the same, at around 0.4 Ah, as shown in Figure 4(b). The major
difference between the two batteries is the initial capacity. The initial capacity
of the Type-B battery reaches 15.0 Ah at cycle zero. However, the duration
of the active zone for the Type-B battery is slightly longer than that of the
Type-A battery. The capacity at the boundary between the active zone and
the steady zone is about 14.1 Ah and 14.3 Ah for the Type-A and Type-B
batteries, respectively.
To calculate the accuracy of the fitted equation for the Type-A and Type-B
batteries we use the mean absolute percentage error between the experimental
3 Results and discussion
C300
Figure 4: (a) The experimental curves for the Type-A and Type-B batteries
and (b) the difference in capacity between the Type-A and Type-B batteries.
1 5 .0
(a )
1 4 .8
T y p e -A
T y p e -B
C a p a c ity (A h )
1 4 .6
1 4 .4
1 4 .2
1 4 .0
1 3 .8
1 3 .6
1 3 .4
1 3 .2
1 3 .0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
N u m b e r s o f C y c le s
2 .0
(b )
1 .8
D iffe r e n c e
C a p a c ity (A h )
1 .6
1 .4
1 .2
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
N u m b e r s o f C y c le s
6 0 0
7 0 0
8 0 0
4 Conclusion
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Table 2: The mape and error values for the two tested batteries.
Category mape (%) Maximum error (%) Minimum error (%)
Type-A
0.18
0.55
0.00068
Type-B
0.16
0.41
0.00042
data and the fitted curve
mape(%) =
1 X |Qreal − Qfit |
× 100% ,
N
Qreal
(2)
where Qreal is the experimental data, Qfit is the fitted curve, N is the total
amount of data recorded for 800 cycles, and the sum is over these 800 data
points.
Table 2 shows the mape value between the experimental data and the fitted
curve is less than 0.18%, and the maximum error is less than 0.55%. Therefore,
the fitted curve closely describes the relationship between the battery capacity
and the number of cycles. In addition, once the battery capacity goes beyond
the steady state, the capacity starts to rapidly decay. Equations (1) and (2)
are used to analyze the capacity conditions of the battery. If the number of
cycles passes more than 2000, and the error rate is much higher than the
maximum error and continues to increase as the number of cycles increases,
then it is an indication that either the limits of the battery capacity have been
reached, or that the battery is fully discharged, meaning that the battery
should be charged.
4
Conclusion
In this study, by separating the exponential and polynomial parts of the fitted
equation, we are able to distinguish variations in the capacity of LiFePO4
batteries corresponding to the different numbers of charge and discharge cycles.
When the battery is in the active state, the capacity decays exponentially,
References
C302
meaning that the exponential term in equation (1) dominates the capacity
behavior. Once the battery reaches the steady zone, it shows a linear decay
in capacity and the polynomial term is dominant. An analysis of the mape
for the tested batteries shows that it is less than 0.18%, meaning that this
fitted curve is closely responsive to the capacity data for the batteries when
the number of cycles is in the range of 0 to 800. Furthermore, if the cycle
number is beyond 800, then equations (1) and (2) can be used to analyze the
capacity conditions.
Acknowledgements The authors acknowledge the support of Ministry of
Science and Technology Taiwan under Grants MOST 104-2221-E-002-108,
104-3113-E-002-004 and 104-3113-E-002-020-CC2.
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Author addresses
1. Chi-Yao Chung, Department of Engineering Science and Ocean
Engineering, National Taiwan University, Taipei 10617, Taiwan.
mailto:[email protected]
2. Kung-Yen Lee, Department of Engineering Science and Ocean
Engineering, National Taiwan University, Taipei 10617, Taiwan.
mailto:[email protected]
3. Ting-Jung Kuo, Department of Engineering Science and Ocean
Engineering, National Taiwan University, Taipei 10617, Taiwan.
mailto:[email protected]
4. Zih-Yeh Lin, Department of Engineering Science and Ocean
Engineering, National Taiwan University, Taipei 10617, Taiwan.
mailto:[email protected]
5. Shuen-De Wu, Department of Mechatronic Engineering, National
Taiwan Normal University, Taipei 10610, Taiwan.
mailto:[email protected]
6. Chiao Fu, Department of Engineering Science and Ocean Engineering,
National Taiwan University, Taipei 10617, Taiwan.
mailto:[email protected]